Answer is: option3
\( 2[F(5) - F(1)] \)Solution:
We are given:
- \( F'(x) = f(x) \)
- We are to evaluate: \( \int_2^{10} f\left( \frac{1}{2}x \right) dx \)
Let:
\( u = \frac{1}{2}x \Rightarrow x = 2u \Rightarrow dx = 2\,du \)
Change the limits:
- When \( x = 2 \), \( u = 1 \)
- When \( x = 10 \), \( u = 5 \)
Now the integral becomes:
\[ \int_2^{10} f\left( \frac{1}{2}x \right) dx = \int_1^5 f(u) \cdot 2\,du = 2 \int_1^5 f(u)\,du \]
Since \( F'(x) = f(x) \), by the Fundamental Theorem of Calculus:
\[ \int_1^5 f(u)\,du = F(5) - F(1) \]
So:
\[ 2 \int_1^5 f(u)\,du = 2[F(5) - F(1)] \]
Hence option C